L. Bers, see section 6.5 of [1]}, showed that an element of \(\Gamma _g\)  is pseudo-Anosov if and only if its action on \(\mathcal {T}_g\)  preserves a (real) geodesic with respect to the Teichmüller, or equivalently Kobayashi, metric. By construction, \(\widetilde{\gamma }\)  is a (real) geodesic with respect to the Kobayashi metric which is preserved by \(\varphi _{\gamma }\) . It follows that \(\varphi _{\gamma }\)  is a pseudo-Anosov element in \(\Gamma _g\) . In this way, to any closed geodesic in \(V\)  we can associate a pseudo-Anosov element in \(\Gamma _g\) . Moreover, if the generators of \(\operatorname{Stab}(\widetilde{\phi }_{(X, q)})\)  are known, then any pseudo-Anosov associated to a closed geodesic in \(V\)  can be written as a unique word (up to conjugation) in these generators.
